Violin acoustics

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Violin acoustics is an area of study within musical acoustics concerned with how the sound of a violin is created as the result of interactions between its many parts. These acoustic qualities are similar to those of other members of the violin family, such as the viola.

The energy of a

During the nineteenth century, the multi-harmonic sound from a bowed string was first studied in detail by the French physicist Félix Savart.^[1][6] The German physicist Hermann von Helmholtz investigated the physics of the plucked string,^[7] and showed that the bowed string travelled in a triangular shape with the apex moving at a constant speed.^[8]

The violin's

A vibrating string does not produce a single frequency. The sound may be described as a combination of a

When the violinist is directed to pluck a string (Ital. pizzicato), the sound produced dies away, or dampens, quickly: the dampening is more striking for a violin compared with the other members of the violin family because of its smaller dimensions, and the effect is greater if an open string is plucked.^[24] During a pizzicato note, the decaying higher harmonics diminish more quickly than the lower ones.^[25]

The

where E is the Young's modulus, S is the cross-sectional area, ΔL is the extension, and L is the string length. For vibrations with a large amplitude, the tension is not constant.^[28] Increasing the tension on a string results in a higher frequency note:

where f is the fundamental frequency of the string, T is the tension force and M is the mass.[14]

The strings of a violin are attached to adjustable tuning pegs and (with some strings) finer tuners. Tuning each string is done by loosening or tightening it until the desired pitch is reached.^[29] The tension of a violin string ranges from 8.7 to 18.7 pounds-force (39 to 83 N).^[30]

Length

For any wave travelling at a speed v, travelling a distance λ in one period T,

${\displaystyle v={{\lambda } \over T}}$.

For a frequency f

${\displaystyle f={\frac {1}{T}}={\frac {v}{\lambda }}}$

For the fundamental frequency of a vibrating string on a violin, the string length is 1/2λ, where λ is the associated wavelength, so

${\displaystyle f={\frac {v}{2L}}}$.^[14]

Materials

String material influences the overtone mix and affects the quality of the sound.^[31] Response and ease of articulation are also affected by choice of string materials.^[31]

Violin strings were originally made from catgut, which is still available and used by some professional musicians,^[32] although strings made of other materials are less expensive to make and are not as sensitive to temperature.^[31] Modern strings are made of steel-core, stranded steel-core, or a synthetic material such as Perlon.^[31] Violin strings (with the exception of most E strings) are helically wound with metal chosen for its density and cost. The winding on a string increases the mass of the string, alters the tone (quality of sound produced) to make it sound brighter or warmer, and affects the response.^[33] A plucked steel string sounds duller than one made of gut, as the action does not deform steel into a pointed shape as easily, and so does not produce as many higher frequency harmonics.^[25]

The bridge

The bridge, which is placed on the top of the body of the violin where the soundboard is highest,^[34] supports one end of the strings' playing length. The static forces acting on the bridge are large, and dependent on the tension in the strings:^[35] 20 lb_f (89 N) passes down through the bridge as a result of a tension in the strings of 50 lb_f (220 N).^[36] The string 'break' angle made by the string across the bridge affects the downward force, and is typically 13 to 15° to the horizontal.^[37]

The bridge transfers energy from the strings to the body of the violin.^[35] As a first approximation, it is considered to act as a node, as otherwise the fundamental frequencies and their related harmonics would not be sustained when a note is played, but its motion is critical in determining how energy is transmitted from the strings to the body, and the behaviour of the strings themselves.^[13] One component of its motion is side-to-side rocking as it moves with the string.^[38] It may be usefully viewed as a mechanical filter, or an arrangement of masses and "springs" that filters and shapes the timbre of the sound.^[39] The bridge is shaped to emphasize a singer's formant at about 3000 Hz.^[40]

Since the early 1980s it has been known that high quality violins have vibrated better at frequencies around 2–3 kHz because of an effect attributed to the resonance properties of the bridge, and now referred as the 'bridge-hill' effect.^[39]

Muting is achieved by fitting a clip onto the bridge, which absorbs a proportion of the energy transmitted to the body of the instrument. Both a reduction in sound intensity and a different timbre are produced, so that using a mute is not seen by musicians as the main method to use when wanting to play more quietly.^[41]

The bow

A violin can sustain its tone by the process of bowing, when friction causes the string to be pulled sideways by the

Violinists generally bow between the bridge and the fingerboard, and are trained to keep the bow perpendicular to the string. In bowing, the three most prominent factors under the player's immediate control are bow speed, force, and the place where the hair crosses the string (known as the 'sounding point'): a vibrating string with a shorter length causes the sounding point to be positioned closer to the bridge. The player may also vary the amount of hair in contact with the string, by tilting the bow stick more or less away from the bridge.^[48] The string twists as it is bowed, which adds a 'ripple' to the waveform: this effect is increased if the string is more massive.^[49]

Bowing directly above the fingerboard (Ital. sulla tastiera) produces what the 20th century American composer and author Walter Piston described as a "very soft, floating quality", caused by the string being forced to vibrate with a greater amplitude.^[50] Sul ponticello—when the bow is played close to the bridge—is the opposite technique, and produces what Piston described as a "glassy and metallic" sound, due to normally unheard harmonics becoming able to affect the timbre.^[51]

Helmholtz motion

Modern research on the physics of violins began with Helmholtz, who showed that the shape of the string as it is bowed is in the form of a 'V', with an apex (known as the 'Helmholtz corner') that moves along the main part of the string at a constant speed. Here, the nature of the friction between bow and string changes, and slipping or sticking occurs, depending on the direction the corner is moving.^[44][52] The wave produced rotates as the Helmholtz corner moves along a plucked string, which caused a reduced amount of energy to be transmitted to the bridge when the plane of rotation is not parallel to the fingerboard. Less energy still is supplied when the string is bowed, as a bow tends to dampen any oscillations that are at an angle to the bow hair, an effect enhanced if an uneven bow pressure is applied, e.g. by a novice player.^[25]

The

Helmholtz and Raman produced models that included sharp cornered waves: the study of smoother corners was undertaken by Cremer and Lazarus in 1968, who showed that significant smoothing occurs (i.e. there are fewer harmonics present) only when normal bowing forces are applied. The theory was further developed during the 1970s and 1980s to produce a digital waveguide model, based on the complex relationship behaviour of the bow's velocity and the frictional forces that were present.^[54] The model was a success in simulating Helmholtz motion (including the 'flattening' effect of the motion caused by larger forces), and was later extended to take into account the string's bending stiffness, its twisting motion, and the effect on the string of body vibrations and the distortion of the bow hair.^[55] However, the model assumed that the coefficient of friction due to the rosin was solely determined by the bow's speed, and ignored the possibility that the coefficient could depend on other variables. By the early 2000s, the importance of variables such the energy supplied by friction to the rosin on the bow, and the player's input into the action of the bow were recognised, showing the need for an improved model.^[56]

The body

The body of a violin is oval and hollow, and has two f-shaped holes, called sound holes, located on either side of the bridge.^[57] The body must be strong enough to support the tension from the strings, but also light and thin enough to vibrate properly.^[36] It is made of two arched wooden plates known as the belly and the backplate, whose sides are formed by thin curved ribs. It acts as a sound box to couple the vibration of strings to the surrounding air, making it audible. In comparison, the strings, which move almost no air, are silent.^[16][58]

The existence of expensive violins is dependent on small differences in their physical behaviour in comparison with cheaper ones.^[59] Their construction, and especially the arching of the belly and the backplate, has a profound effect on the overall sound quality of the instrument,^[60] and its many different resonant frequencies are caused by the nature of the wooden structure. The different parts all respond differently to the notes that are played, displaying what Carleen Hutchins described as 'wood resonances'.^[1] The response of the string can be tested by detecting the motion produced by the current through a metal string when it is placed in an oscillating magnetic field.^[13] Such tests have shown that the optimum 'main wood resonance' (the wood resonance with the lowest frequency) occurs between 392 and 494 Hz, equivalent to a tone below and above A₄.^[61]

The ribs are reinforced at their edges with lining strips, which provide extra gluing surface where the plates are attached.^[36] The wooden structure is filled, glued and varnished using materials which all contribute to a violin's characteristic sound.^[62] The air in the body also acts to enhance the violin's resonating properties, which are affected by the volume of enclosed air and the size of the f-holes.^[63]

The belly and the backplate can display modes of vibration when they are forced to vibrate at particular frequencies. The many modes that exist can be found using fine dust or sand, sprinkled on the surface of a violin-shaped plate. When a mode is found, the dust accumulates at the (stationary) nodes: elsewhere on the plate, where it is oscillating, the dust fails to appear. The patterns produced are named after the German physicist Ernst Chladni, who first developed this experimental technique.^[16]

Modern research has used sophisticated techniques such as holographic interferometry, which enables analysis of the motion of the violin surface to be measured, a method first developed by scientists in the 1960s, and the finite element method, where discrete parts of the violin are studied with the aim of constructing an accurate simulation. The British physicist Bernard Richardson has built virtual violins using these techniques.^[16] At East Carolina University, the American acoustician George Bissinger has used laser technology to produce frequency responses that have helped him to determine how the efficiency and damping of the violin's vibrations depend on frequency.^[16] Another technique, known as modal analysis, involves the use of 'tonal copies' of old instruments to compare a new instrument with an older one. The effects of changing the new violin in the smallest way can be identified, with the aim of replicating the tonal response of the older model.^[64]

The bass bar and the sound post

A bass bar and a sound post concealed inside the body both help transmit sound to the back of the violin, with the sound post also serving to support the structure. The bass bar is glued to the underside of the top, whilst the sound post is held in place by friction. The bass bar was invented to strengthen the structure, and is positioned directly below one of the bridge's feet.^[36][65] Near the foot of the bridge, but not directly below it, is the sound post.^[66]

When the bridge receives energy from the strings, it rocks, with the sound post acting as a pivot and the bass bar moving with the plate as the result of leverage. This behaviour enhances the violin tone quality: if the sound post's position is adjusted, or if the forces acting on it are changed, the sound produced by the violin can be adversely affected.^[36] Together they make the shape of the violin body asymmetrical, which allows different vibrations to occur, which causing the timbre to become more complex.^[16]

In addition to the normal modes of the body structure, the enclosed air in the body exhibits Helmholtz resonance modes as it vibrates.^[67]

Wolf tones

Bowing is an example of resonance where maximum amplification occurs at the natural frequency of the system, and not the forcing frequency, as the bow has no periodic force.^[68] A wolf tone is produced when small changes in the fundamental frequency—caused by the motion of the bridge—become too great, and the note becomes unstable.^[13] A sharp resonance response from the body of a cello (and occasionally a viola or a violin) produces a wolf tone, an unsatisfactory sound that repeatedly appears and disappears. A correctly positioned suppressor can remove the tone by reducing the resonance at that frequency, without dampening the sound of the instrument at other frequencies.^[69]

Comparison with other members of the violin family

The physics of the viola are the same as that of the violin, and the construction and acoustics of the cello and the double bass are similar.^[70]

The viola is a larger version of the violin, and has on average a total body length of 27+1⁄4 inches (69.2 cm), with strings tuned a fifth lower than a violin (with a length of about 23+3⁄8 inches (59.4 cm)). The viola's larger size is not proportionally great enough to correspond to the strings being pitched as they are, which contributes to its different timbre. Violists need to have hands large enough to be able to accomplish fingering comfortably. The C string has been described by Piston as having a timbre that is "powerful and distinctive",^[71] but perhaps in part because the sound it produces is easily covered, the viola is not so frequently used in the orchestra as a solo instrument.^[72] According to the American physicist John Rigden, the lower notes of the viola (along with the cello and the double bass) suffer from strength and quality. This is because typical resonant frequencies for a viola lie between the natural frequencies of the middle open strings, and are too high to reinforce the frequencies of the lower strings. To correct this problem, Rigden calculated that a viola would need strings that were half as long again as on a violin, which would making the instrument inconvenient to play.^[73]

The cello, with an overall length of 48 inches (121.9 cm), is pitched an octave below the viola. The proportionally greater thickness of its body means that its timbre is not adversely affected by having dimensions that do not correspond to its pitch of its open strings, as is the case with the viola.^[74]

The double bass, in comparison with the other members of the family, is more pointed where the belly is joined by the neck, possibly to compensate for the strain caused by the tension of the strings, and is fitted with cogs for tuning the strings.^[75][76] The average overall length of an orchestral bass is 74 inches (188.0 cm).^[76] The back can be arched or flat. The bassist's fingers have to stretch twice as far as a cellist's, and greater force is required to press them against the finger-board. The pizzicato tone, which is 'rich' sounding due to the slow speed of vibrations, is changeable according to which of the associated harmonies are more dominant. The technical capabilities of the double bass are limited. Quick passages are seldom written for it; they lack clarity because of the time required for the strings to vibrate. The double bass is the foundation of the whole orchestra and therefore musically of great importance.^[75] According to John Rigden, a double bass would need to be twice as large as its present size for its bowed notes to sound powerful enough to be heard over an orchestra.^[77]

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